Imagine the following simulated data

##          
##           better normal
##   failure     50     32
##   success     50     68

It appears that one is better off going to the normal hospital because there is a 68% success rate.

Note that so far, we have a simple story.

We are only measuring the outcome of failure or success and the type of hospital.

Let’s add something to the story.

We now consider severity of case, in addition to type of hospital and outcome.

## , ,  = less severe
## 
##          
##           better normal
##   failure      2     16
##   success     18     64
## 
## , ,  = more severe
## 
##          
##           better normal
##   failure     48     16
##   success     32      4

Considering severity of case, the rate of success is always better in the better hospital.

Visualize it.

The disparity arises because more of the more severe cases go to the better hospital. Once we account for this disparity, we see the the better hospital performs better for both levels of severity.